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Number of partitions of n into parts 7k and 7k+2 with at least one part of each type.
3

%I #15 Aug 16 2020 20:26:42

%S 0,0,0,0,0,0,0,0,1,0,1,0,1,0,1,3,1,3,1,3,1,3,7,3,8,3,8,3,8,14,8,17,8,

%T 18,8,18,26,18,33,18,36,18,37,47,37,61,37,68,37,71,81,72,106,72,121,

%U 72,128,138,131,181,132,209,132,224,228,231,297,234,347,235,376,373

%N Number of partitions of n into parts 7k and 7k+2 with at least one part of each type.

%H Robert Price, <a href="/A035652/b035652.txt">Table of n, a(n) for n = 1..1000</a>

%F G.f. : (-1 + 1/Product_{k>=0} (1 - x^(7 k + 2)))*(-1 + 1/Product_{k>=1} (1 - x^(7 k))). - _Robert Price_, Aug 12 2020

%t nmax = 72; s1 = Range[1, nmax/7]*7; s2 = Range[0, nmax/7]*7 + 2;

%t Table[Count[IntegerPartitions[n, All, s1~Join~s2],

%t x_ /; ContainsAny[x, s1 ] && ContainsAny[x, s2 ]], {n, 1, nmax}] (* _Robert Price_, Aug 12 2020 *)

%t nmax = 72; l = Rest@CoefficientList[Series[(-1 + 1/Product[(1 - x^(7 k)), {k, 1, nmax}])*(-1 + 1/Product[(1 - x^(7 k + 2)), {k, 0, nmax}]), {x, 0, nmax}], x] (* _Robert Price_, Aug 12 2020 *)

%Y Cf. A035441-A035468, A035618-A035651, A035653-A035699.

%K nonn

%O 1,16

%A _Olivier GĂ©rard_